Chapter 3: Techniques of IntegrationContent check — Calculus Vol. 2, Chapter 3: Techniques of Integration

Evaluate ${{\displaystyle \int }}^{\text{}}x{e}^{2x}dx$ using the integration-by-parts formula with $u=x$ and $dv=e^{2x}dx.$

Evaluate ${{\displaystyle \int }}^{\text{}}x\text{ln}xdx.$

Evaluate ${{\displaystyle \int }}^{\text{}}{x}^2\text{sin}xdx.$

Evaluate ${\displaystyle {\int }_0^{\pi \text{/}2}x\text{cos}xdx.}$

In using the technique of integration by parts, you must carefully choose which expression is u. For each of the following problems, use the guidelines in this section to choose u. Do not evaluate the integrals.

${\displaystyle \int x^3{e}^{2x}dx}$

In using the technique of integration by parts, you must carefully choose which expression is u. For each of the following problems, use the guidelines in this section to choose u. Do not evaluate the integrals.

${\displaystyle \int y^3\text{cos}ydy}$

In using the technique of integration by parts, you must carefully choose which expression is u. For each of the following problems, use the guidelines in this section to choose u. Do not evaluate the integrals.

${\displaystyle \int e^{3x}\text{sin}(2x)dx}$

Find the integral by using the simplest method. Not all problems require integration by parts.

${\displaystyle \int \text{ln}xdx}$ (Hint: ${\displaystyle \int \text{ln}xdx}$ is equivalent to ${\displaystyle \int 1·\text{ln}(x)dx}.)$

Find the integral by using the simplest method. Not all problems require integration by parts.

${\displaystyle \int {\text{tan}}^{\mathrm{-1}}xdx}$

Find the integral by using the simplest method. Not all problems require integration by parts.

${\displaystyle \int x\text{sin}(2x)dx}$

Find the integral by using the simplest method. Not all problems require integration by parts.

${\displaystyle \int x{e}^{\text{−}x}dx}$

Find the integral by using the simplest method. Not all problems require integration by parts.

${\displaystyle \int x^2}\text{cos}xdx$

Find the integral by using the simplest method. Not all problems require integration by parts.

${\displaystyle \int \text{ln}(2x+1)dx}$

Find the integral by using the simplest method. Not all problems require integration by parts.

${\displaystyle \int e^x\text{sin}xdx}$

Find the integral by using the simplest method. Not all problems require integration by parts.

${\displaystyle \int x{e}^{\text{−}{x}^2}dx}$

Find the integral by using the simplest method. Not all problems require integration by parts.

${\displaystyle \int \text{sin}(\text{ln}(2x))dx}$

Find the integral by using the simplest method. Not all problems require integration by parts.

${\displaystyle \int {(\text{ln}x)}^{2}dx}$

Find the integral by using the simplest method. Not all problems require integration by parts.

${\displaystyle \int x^2\text{ln}xdx}$

Find the integral by using the simplest method. Not all problems require integration by parts.

${\displaystyle \int {\text{cos}}^{\mathrm{-1}}(2x)dx}$

Find the integral by using the simplest method. Not all problems require integration by parts.

${\displaystyle \int x^2\text{sin}xdx}$

Find the integral by using the simplest method. Not all problems require integration by parts.

${\displaystyle \int x^3\text{sin}xdx}$

Find the integral by using the simplest method. Not all problems require integration by parts.

${\displaystyle \int x{\text{sec}}^{\mathrm{-1}}xdx}$

Find the integral by using the simplest method. Not all problems require integration by parts.

${\displaystyle \int x\text{cosh}xdx}$

Compute the definite integrals. Use a graphing utility to confirm your answers.

${\displaystyle {\int }_0^{1}x{e}^{\mathrm{-2}x}dx}$ (Express the answer in exact form.)

Compute the definite integrals. Use a graphing utility to confirm your answers.

${\displaystyle {\int }_1^e\text{ln}(x^2)dx}$

Compute the definite integrals. Use a graphing utility to confirm your answers.

${\displaystyle {\int }_{\text{−}\pi }^{\pi }x\text{sin}xdx}$ (Express the answer in exact form.)

Compute the definite integrals. Use a graphing utility to confirm your answers.

${\displaystyle {\int }_0^{\pi \text{/}2}{x}^2\text{sin}xdx}$ (Express the answer in exact form.)

Compute the definite integrals. Use a graphing utility to confirm your answers.

Evaluate ${\displaystyle \int \text{cos}x\text{ln}(\text{sin}x)dx}$

Derive the following formulas using the technique of integration by parts. Assume that n is a positive integer. These formulas are called reduction formulas because the exponent in the x term has been reduced by one in each case. The second integral is simpler than the original integral.

${\displaystyle \int x^n\text{cos}xdx}=$ $x^n\text{sin}x-$ $n{\displaystyle \int x^{n-1}\text{sin}xdx}$

Derive the following formulas using the technique of integration by parts. Assume that n is a positive integer. These formulas are called reduction formulas because the exponent in the x term has been reduced by one in each case. The second integral is simpler than the original integral.

Integrate ${\displaystyle \int 2x\sqrt{2x-3}}dx$ using two methods:

a. Using parts, letting $dv=\sqrt{2x-3}dx$

b. Substitution, letting $u=2x-3$

State whether you would use integration by parts to evaluate the integral. If so, identify u and dv. If not, describe the technique used to perform the integration without actually doing the problem.

${\displaystyle \int \frac{{\text{ln}}^{2}x}{x}dx}$

State whether you would use integration by parts to evaluate the integral. If so, identify u and dv. If not, describe the technique used to perform the integration without actually doing the problem.

${\displaystyle \int x{e}^{x^2-3}dx}$

State whether you would use integration by parts to evaluate the integral. If so, identify u and dv. If not, describe the technique used to perform the integration without actually doing the problem.

${\displaystyle \int x^2\text{sin}(3x^3+2)dx}$

Sketch the region bounded above by the curve, the x-axis, and $x=1,$ and find the area of the region. Provide the exact form or round answers to the number of places indicated.

$y=$ $e^{\text{−}x}\text{sin}(\pi x)$ (Approximate answer to five decimal places.)

Find the volume generated by rotating the region bounded by the given curves about the specified line. Express the answers in exact form or approximate to the number of decimal places indicated.

$y=e^{\text{−}x}$ $y=$ $0,$ $x=$ $\mathrm{-1}x=$ $0;$ about $x=1$ (Express the answer in exact form.)

Find the volume generated by rotating the region bounded by the given curves about the specified line. Express the answers in exact form or approximate to the number of decimal places indicated.

Find the area under the graph of $y={\text{sec}}^{3}x$ from $x=0\text{to}x=1.$ (Round the answer to two significant digits.)

Find the volume generated by rotating the region bounded by the given curves about the specified line. Express the answers in exact form or approximate to the number of decimal places indicated.

Find the area of the region enclosed by the curve $y=x\text{cos}x$ and the x-axis for

$\frac{11\pi }{2}\le x\le \frac{13\pi }{2}.$ (Express the answer in exact form.)

Find the volume generated by rotating the region bounded by the given curves about the specified line. Express the answers in exact form or approximate to the number of decimal places indicated.

Find the volume of the solid generated by revolving the region bounded by the curve $y=4\text{cos}x$ and the x-axis, $\frac{\pi }{2}\le x\le \frac{3\pi }{2},$ about the x-axis. (Express the answer in exact form.)

Evaluate ${\displaystyle \int }{\text{sin}}^{4}x\text{cos}xdx.$

Evaluate ${\displaystyle \int }{\text{cos}}^{3}x{\text{sin}}^{2}xdx.$

Evaluate ${\displaystyle \int {\text{cos}}^{2}xdx.}$

Evaluate ${\displaystyle \int }{\text{cos}}^{3}xdx.$

Evaluate ${\displaystyle \int }{\text{cos}}^2(3x)dx.$

Evaluate ${\displaystyle \int }\text{cos}\left(6x\right)\text{cos}(5x)dx.$

Evaluate ${\displaystyle \int }{\text{tan}}^{5}x{\text{sec}}^{2}xdx.$

Evaluate ${\displaystyle \int }{\text{tan}}^{3}x{\text{sec}}^{7}xdx.$

Apply the reduction formula to ${\displaystyle \int }{\text{sec}}^{5}xdx.$

Fill in the blank to make a true statement.

${\text{sin}}^{2}x+$ $\_\_\_\_\_\_\_=$ $1$

Use an identity to reduce the power of the trigonometric function to a trigonometric function raised to the first power.

${\text{sin}}^{2}x=$ $\_\_\_\_\_\_\_$

Evaluate each of the following integrals by u-substitution.

${\displaystyle \int {\text{sin}}^{3}x\text{cos}xdx}$

Evaluate each of the following integrals by u-substitution.

${\displaystyle \int {\text{tan}}^5(2x){\text{sec}}^2(2x)dx}$

Evaluate each of the following integrals by u-substitution.

${\displaystyle \int \text{tan}\left(\frac{x}{2}\right){\text{sec}}^2\left(\frac{x}{2}\right)dx}$

Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may be done using techniques of integration learned previously.)

${\displaystyle \int {\text{sin}}^{3}xdx}$

Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may be done using techniques of integration learned previously.)

${\displaystyle \int \text{sin}x\text{cos}xdx}$

Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may be done using techniques of integration learned previously.)

${\displaystyle \int {\text{sin}}^{5}x{\text{cos}}^{2}xdx}$

Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may be done using techniques of integration learned previously.)

${\displaystyle \int \sqrt{\text{sin}x}}\text{cos}xdx$

Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may be done using techniques of integration learned previously.)

${\displaystyle \int \text{sec}x\text{tan}xdx}$

Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may be done using techniques of integration learned previously.)

${\displaystyle \int {\text{tan}}^{2}x\text{sec}xdx}$

Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may be done using techniques of integration learned previously.)

${\displaystyle \int {\text{sec}}^{4}xdx}$

Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may be done using techniques of integration learned previously.)

${\displaystyle \int \text{csc}xdx}$

For the following exercises, find a general formula for the integrals.

${\displaystyle \int {\text{sin}}^{2}ax\text{cos}axdx}$

Use the double-angle formulas to evaluate the following integrals.

${\displaystyle {\int }_0^{\pi }{\text{sin}}^{2}xdx}$

Use the double-angle formulas to evaluate the following integrals.

${\displaystyle \int {\text{cos}}^{2}3xdx}$

Use the double-angle formulas to evaluate the following integrals.

${\displaystyle \int {\text{sin}}^{2}xdx}+$ ${\displaystyle \int {\text{cos}}^{2}xdx}$

For the following exercises, evaluate the definite integrals. Express answers in exact form whenever possible.

${\displaystyle {\int }_0^{2\pi }\text{cos}x\text{sin}2xdx}$

For the following exercises, evaluate the definite integrals. Express answers in exact form whenever possible.

${\displaystyle {\int }_0^{\pi }\text{cos}(99x)\text{sin}(101x)dx}$

For the following exercises, evaluate the definite integrals. Express answers in exact form whenever possible.

${\displaystyle {\int }_0^{2\pi }\text{sin}x\text{sin}(2x)\text{sin}(3x)dx}$

For the following exercises, evaluate the definite integrals. Express answers in exact form whenever possible.

${\displaystyle {\int }_{\pi \text{/}6}^{\pi \text{/}3}\frac{{\text{cos}}^{3}x}{\sqrt{\text{sin}x}}dx}$ (Round this answer to three decimal places.)

For the following exercises, evaluate the definite integrals. Express answers in exact form whenever possible.

${\displaystyle {\int }_0^{\pi \text{/}2}\sqrt{1-\text{cos}(2x)}dx}$

For the following exercises, evaluate the definite integrals. Express answers in exact form whenever possible.

Find the area of the region bounded by the graphs of the equations $y=$ ${\text{cos}}^{2}x,$ $y=$ ${\text{sin}}^{2}x,$ $x=$ $-$ $\frac{\pi }{4},$ $\text{and}x=$ $\frac{\pi }{4}.$

For the following exercises, evaluate the definite integrals. Express answers in exact form whenever possible.

Find the average value of the function $f(x)=$ ${\text{sin}}^{2}x{\text{cos}}^{3}x$ over the interval $\left[\text{−}\pi ,\pi \right].$

For the following exercises, solve the differential equations.

$\frac{dy}{d\theta }=$ ${\text{sin}}^4\left(\pi \theta \right)$

For the following exercises, solve the differential equations.

Find the length of the curve $y=$ $\text{ln}(\text{sin}x),$ $\frac{\pi }{3}\le x\le \frac{\pi }{2}.$

For the following exercises, use this information: The inner product of two functions f and g over $[a,b]$ is defined by $f(x)·g(x)=$ $\langle f,$ $g\rangle =$ ${\displaystyle {\int }_a^{b}f·gdx.}$ Two distinct functions f and g are said to be orthogonal if $\langle f,g\rangle =0.$

Show that $\{\text{sin}(2x),\text{cos}(3x)\}$ are orthogonal over the interval $\left[\text{−}\pi ,\pi \right].$

For the following exercises, use this information: The inner product of two functions f and g over $[a,b]$ is defined by $f(x)·g(x)=$ $\langle f,$ $g\rangle =$ ${\displaystyle {\int }_a^{b}f·gdx.}$ Two distinct functions f and g are said to be orthogonal if $\langle f,g\rangle =0.$

Integrate $y^{\prime }=$ $\sqrt{\text{tan}x}{\text{sec}}^{4}x.$

For each pair of integrals, determine which one is more difficult to evaluate. Explain your reasoning.

${\displaystyle \int {\text{tan}}^{350}x{\text{sec}}^{2}xdx}$ or ${\displaystyle \int {\text{tan}}^{350}x\text{sec}xdx}$

Rewrite the integral ${\displaystyle \int \frac{x^3}{\sqrt{25-x^2}}dx}$ using the appropriate trigonometric substitution (do not evaluate the integral).

Rewrite ${{\displaystyle \int }}^{\text{}}{x}^3\sqrt{x^2+4}dx$ by using a substitution involving $\text{tan}\theta .$

Evaluate ${\displaystyle \int \frac{dx}{\sqrt{x^2-4}}}.$ Assume that $x>2.$

Simplify the following expressions by writing each one using a single trigonometric function.

$9{\text{sec}}^2\theta -9$

Simplify the following expressions by writing each one using a single trigonometric function.

$a^2+a^2{\text{sinh}}^2\theta$

Use the technique of completing the square to express each trinomial as the square of a binomial or the square of a binomial plus a constant.

$4x^2-4x+1$

Use the technique of completing the square to express each trinomial as the square of a binomial or the square of a binomial plus a constant.

$\text{−}{x}^2-2x+4$

Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable.

${\displaystyle \int \frac{dx}{\sqrt{x^2-a^2}}}$

Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable.

${\displaystyle \int \frac{dx}{\sqrt{1+9x^2}}}$

Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable.

${\displaystyle \int \frac{dx}{x^2\sqrt{1-x^2}}}$

Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable.

${\displaystyle \int \sqrt{x^2+9}}dx$

Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable.

${\displaystyle \int \frac{{\theta }^{3}d\theta }{\sqrt{9-{\theta }^2}}}$

Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable.

${\displaystyle \int \sqrt{x^6-x^8}}dx$

Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable.

${\displaystyle \int \frac{dx}{{\left(x^2-9\right)}^{3\text{/}2}}}$

Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable.

${\displaystyle \int \frac{x^{2}dx}{\sqrt{x^2-1}}}$

Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable.

${\displaystyle \int \frac{dx}{x^2\sqrt{x^2+1}}}$

Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable.

$\int {(1-x^2)}^{3\text{/}2}dx$

In the following exercises, use the substitutions $x=$ $\text{sinh}\theta ,$ $\text{cosh}\theta ,$ or $\text{tanh}\theta .$ Express the final answers in terms of the variable x.

${\displaystyle \int \frac{dx}{x\sqrt{1-x^2}}}$

In the following exercises, use the substitutions $x=$ $\text{sinh}\theta ,$ $\text{cosh}\theta ,$ or $\text{tanh}\theta .$ Express the final answers in terms of the variable x.

${\displaystyle \int \frac{\sqrt{x^2-1}}{x^2}dx}$

In the following exercises, use the substitutions $x=$ $\text{sinh}\theta ,$ $\text{cosh}\theta ,$ or $\text{tanh}\theta .$ Express the final answers in terms of the variable x.

${\displaystyle \int \frac{\sqrt{1+x^2}}{x^2}dx}$

Use the technique of completing the square to evaluate the following integrals.

${\displaystyle \int \frac{1}{x^2+2x+1}dx}$

Use the technique of completing the square to evaluate the following integrals.

${\displaystyle \int \frac{1}{\sqrt{\text{−}{x}^2+10x}}dx}$

Use the technique of completing the square to evaluate the following integrals.

Evaluate the integral without using calculus: ${\displaystyle {\int }_{\mathrm{-3}}^3\sqrt{9-x^2}dx.}$

Use the technique of completing the square to evaluate the following integrals.

Evaluate the integral ${\displaystyle \int \frac{dx}{\sqrt{1-x^2}}}$ using two different substitutions. First, let $x=\text{cos}\theta$ and evaluate using trigonometric substitution. Second, let $x=\text{sin}\theta$ and use trigonometric substitution. Are the answers the same?

Use the technique of completing the square to evaluate the following integrals.

Evaluate the integral ${\displaystyle \int \frac{x}{x^2+1}}dx$ using the form ${\displaystyle \int \frac{1}{u}du}.$ Next, evaluate the same integral using $x=\text{tan}\theta .$ Are the results the same?

Use the technique of completing the square to evaluate the following integrals.

State the method of integration you would use to evaluate the integral ${\displaystyle \int x^2\sqrt{x^2-1}dx}.$ Why did you choose this method?

Use the technique of completing the square to evaluate the following integrals.

Find the length of the arc of the curve over the specified interval: $y=$ $\text{ln}x,$ $[1,5].$ Round the answer to three decimal places.

Use the technique of completing the square to evaluate the following integrals.

The region bounded by the graph of $f(x)=\frac{1}{1+x^2}$ and the x-axis between $x=0$ and $x=1$ is revolved about the x-axis. Find the volume of the solid that is generated.

Solve the initial-value problem for y as a function of x.

$(64-x^2)\frac{dy}{dx}=$ $1,$ $y(0)=$ $3$

Solve the initial-value problem for y as a function of x.

An oil storage tank can be described as the volume generated by revolving the area bounded by $y=$ $\frac{16}{\sqrt{64+x^2}},$ $x=$ $0,$ $y=$ $0,$ $x=$ $2$ about the x-axis. Find the volume of the tank (in cubic meters).

Solve the initial-value problem for y as a function of x.

Find the length of the curve $y=\sqrt{16-x^2}$ between $x=0$ and $x=2.$

Evaluate ${\displaystyle \int \frac{x-3}{x+2}dx.}$

Evaluate ${\displaystyle \int \frac{x+1}{(x+3)(x-2)}dx}.$

Set up the partial fraction decomposition for ${\displaystyle \int \frac{x+2}{{\left(x+3\right)}^3{(x-4)}^2}dx.}$ (Do not solve for the coefficients or complete the integration.)

Set up the partial fraction decomposition for ${\displaystyle \int \frac{x^2+3x+1}{(x+2){(x-3)}^2{(x^2+4)}^2}dx.}$

Express the rational function as a sum or difference of two simpler rational expressions.

$\frac{x^2+1}{x(x+1)(x+2)}$

Express the rational function as a sum or difference of two simpler rational expressions.

$\frac{3x+1}{x^2}$

Express the rational function as a sum or difference of two simpler rational expressions.

$\frac{2x^4}{x^2-2x}$

Express the rational function as a sum or difference of two simpler rational expressions.

$\frac{1}{x^2(x-1)}$

Express the rational function as a sum or difference of two simpler rational expressions.

$\frac{1}{x(x-1)(x-2)(x-3)}$

Express the rational function as a sum or difference of two simpler rational expressions.

$\frac{3x^2}{x^3-1}=$ $\frac{3x^2}{(x-1)(x^2+x+1)}$

Express the rational function as a sum or difference of two simpler rational expressions.

$\frac{3x^4+x^3+20x^2+3x+31}{(x+1){\left(x^2+4\right)}^2}$

Use the method of partial fractions to evaluate each of the following integrals.

${\displaystyle \int \frac{3x}{x^2+2x-8}dx}$

Use the method of partial fractions to evaluate each of the following integrals.

${\displaystyle \int \frac{x}{x^2-4}dx}$

Use the method of partial fractions to evaluate each of the following integrals.

${\displaystyle \int \frac{2x^2+4x+22}{x^2+2x+10}dx}$

Use the method of partial fractions to evaluate each of the following integrals.

${\displaystyle \int \frac{2-x}{x^2+x}dx}$

Use the method of partial fractions to evaluate each of the following integrals.

${\displaystyle \int \frac{dx}{x^3-2x^2-4x+8}}$

Evaluate the following integrals, which have irreducible quadratic factors.

${\displaystyle \int \frac{2}{(x-4)\left(x^2+2x+6\right)}dx}$

Evaluate the following integrals, which have irreducible quadratic factors.

${\displaystyle \int \frac{x^3+6x^2+3x+6}{x^3+2x^2}dx}$

Use the method of partial fractions to evaluate the following integrals.

${\displaystyle \int \frac{3x+4}{\left(x^2+4\right)(3-x)}dx}$

Use the method of partial fractions to evaluate the following integrals.

${\displaystyle \int \frac{3x+4}{x^3-2x-4}dx}$ (Hint: Use the rational root theorem.)

Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals.

${\displaystyle \int \frac{e^x}{e^{2x}-e^x}dx}$